On 5/6/2019 12:44 AM, Bruce Kellett wrote:
On Mon, May 6, 2019 at 4:41 PM Jason Resch <[email protected]
<mailto:[email protected]>> wrote:
On Mon, May 6, 2019 at 1:19 AM Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
This is essentially the point that both Turing and Goedel made
when they pointed out that human consciousness is not Turing
emulable -- it involves intuitive leaps that are not
algorithmic, presumable coming from an uncodable environment.
Could you provide citations to Turing and Godel's thoughts on
this? In my view Turing was the founder of
functionalism/computationalism, when in his 1950 paper "Computing
Machinery and Intelligence" he wrote:
“The fact that Babbage's Analytical Engine
was to be entirely mechanical will help us rid ourselves of a
superstition. Importance is often
attached to the fact that modern digital computers are
electrical, and the nervous system is also
electrical. Since Babbage's machine was not electrical, and
since all digital computers are in a sense
equivalent, we see that this use of electricity cannot be of
theoretical importance. [...] If we wish to
find such similarities we should look rather for mathematical
analogies of function.”
As for Godel, while I am aware of instances where his ideas have
been misapplied by some philosophers to argue that human
consciousness is not Turing emulable, I am not aware of any
writings of Godel where he expressed such ideas. It is hard for me
to believe Godel himself misunderstood his own ideas to the extent
necessary to believe human mathematicians somehow immune to its
implications. According to Godel's 14 points (his own personal
philosophy) it suggests he sees nothing special about the material
composition, and he also believes all problems (including art) can
be addressed through systematic methods. This suggests to me he
would be a proponent of at least "weak AI", which again is
sufficient for my thought experiment.
1. The world is rational.
2. Human reason can, in principle, be developed more highly
(through certain techniques).
*3. There are systematic methods for the solution of all
problems (also art, etc.).*
*4. There are other worlds and rational beings of a different
and higher kind.*
5. The world in which we live is not the only one in which we
shall live or have lived.
6. There is incomparably more knowable a priori than is
currently known.
7. The development of human thought since the Renaissance is
thoroughly intelligible (durchaus einsichtige).
8. Reason in mankind will be developed in every direction.
9. Formal rights comprise a real science.
*10. Materialism is false.*
*11. The higher beings are connected to the others by analogy,
not by composition.*
12. Concepts have an objective existence.
13. There is a scientific (exact) philosophy and theology,
which deals with concepts of the highest abstractness; and
this is also most highly fruitful for science.
14. Religions are, for the most part, bad– but religion is not.
(Emphasis mine)
Jason
I base these comments on an analysis in a paper by Copeland and
Shagrir, in the book "Computability: Turing, Goedel, Church, and
Beyond" (MIT Press, 2015). The main argument is that "In about 1970,
Goedel wrote a brief note entitled 'A Philosophical Error in Turing's
Work' (1972; in Goedel's Collected Works)." "In the postscript, Goedel
also raised the intriguing 'question of whether there exist finite
non-mechanical procedures'; and he observed that the generalised
incompleteness results 'do not establish any bounds for the powers of
human reason, but rather for the potentialities of pure formalism in
mathematics."
"A philosophical error in Turing's work. Turing in [section 9 of "On
Computable Numbers" (1936, 75-76)} gives an argument which is supposed
to show that mental procedures cannot go beyond mechanical procedures.
However ... what Turing disregards completely is the fact that mind,
in its use, is not static, but constantly developing ... Although at
each stage the number and precision of the abstract terms at our
disposal may be finite, both (and, therefore, also Turing's number of
distinguishable states of mind) may converge toward infinity in the
course of the application of the procedure. (Geode 1972, 306)."
Further: "What Turing disregards completely is the fact that mind, in
its use, is not static, but constantly developing. This is seen, e.g.,
from the infinite series of ever stronger axioms of infinity in set
theory, each of which expresses a new idea or insight ... Therefore,
although at each stage of the mind's development the number of
possible states is finite, there is no reason why this number should
not converge to infinity in the course of its development. (Godel in
Wang 1974, 325)."
There's a very good reason this number will not "converge to infinity".
The brain can only change at a limited a rate and as Conrad said, ""Art
is long and life is short, and success is very far off." The space of
possible TM's instantiated by human brains, even allowing for
modification by the envrionment, is still finite.
The article by Copeland and Shagrir then goes on to defend Turing
against Goedel's criticism, by pointing out that Turing actually says
"Having defined a certain infinite binary sequence \delta, which he
shows to be uncomputable, Turing says: "It is (so far as we know at
present) possible that any assigned number of figures of \delta can be
calculated, but not by a uniform process. When sufficiently many
figures of \delta have been calculated, an essentially new method is
necessary in order to obtain more figures". This sequence of
essentially new methods is, itself, uncomputable.
Isn't this just his point that no single Turing machine can solve the
halting problem for every TM, even though it may be able to solve the
problem for many TMs.
Brent
In Turing's view, the activity of what he called the faculty of
intuition brings it about that mathematical judgments exceed what can
be expressed by means of a single formal system.
I recommend going to the original Copeland and Shagrir paper for more
detail.
Bruce
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